# Why does cos(22.5°) not equal 2/√5?

• bonodut
In summary, the conversation discusses a misunderstanding about the relationship between angles and slopes in trigonometry. The individual believes that halving an angle should result in a doubling of the base leg, but this is incorrect. The correct angle for a right triangle with sides 1, 2, and √5 is approximately 26.5 degrees, not 22.5 degrees. The cosine of this angle is 2/√5, which is verified by online calculators.
bonodut
I've never really learned any trigonometry and I'm wondering if someone could tell me where I'm going wrong here.

My reasoning is this:

The hypotenuse of a 1,1,√2 right triangle is at a 45° angle to its base.

Halving that angle should require that the base leg be doubled if the height leg is kept at 1, giving the new triangle lengths of 1,2,√5

The cosine is adjacent/hypotenuse which would be 2/√5, yet calculators give a different answer. Why is this?

bonodut said:
Halving that angle should require that the base leg be doubled if the height leg is kept at 1, giving the new triangle lengths of 1,2,√5
Your mistake is in thinking that a 1, 2, sqrt(5) triangle has one angle 22.5 degrees.

phinds said:
Your mistake is in thinking that a 1, 2, sqrt(5) triangle has one angle 22.5 degrees.
bonodut said:
Halving that angle should require that the base leg be doubled if the height leg is kept at 1

So this assumption isn't correct?

Edit: I see now that it's not but I don't understand why... Seems like it should be

2nd edit: my problem was that I was equating angle with slope. I like math... Too bad I'm so bad at it! :/

Last edited:
The relation between angle and slope is

$$\mathrm{slope}=\dfrac{\mathrm{rise}}{\mathrm{run}}=\dfrac{\Delta y}{\Delta x}=\dfrac{\sin(\theta)}{\cos(\theta)}=\tan(\theta)$$

ellipsis
bonodut said:
So this assumption isn't correct?

Edit: I see now that it's not but I don't understand why... Seems like it should be

2nd edit: my problem was that I was equating angle with slope. I like math... Too bad I'm so bad at it! :/
Draw a right angle triangle using protractor,scale,pen or pencil which I just did.By drawing the dimensions of right angle triangle of sides 1,2,## \sqrt 5 ## you will see that the angle made by hypotenuse and base is approximately 26.5° not your 22.5°.So, cos 26.5°= 2/##\sqrt 5## So the calculators were right and you may have made a minute mistake.

## 1. Why isn't cos(22.5°) equal to 2/√5?

Cosine is a trigonometric function that represents the ratio of the adjacent side to the hypotenuse in a right triangle. As such, it can only take on values between -1 and 1. The value of 2/√5 is approximately 0.8944, which is outside of this range. Therefore, it is not possible for cos(22.5°) to be equal to 2/√5.

## 2. Can cos(22.5°) be simplified to a value closer to 2/√5?

No, cos(22.5°) cannot be simplified further. It is already in its simplest form and cannot be expressed as a fraction with integers in the numerator and denominator.

## 3. Why does the calculator give a different value for cos(22.5°) than 2/√5?

Calculators use algorithms and approximations to calculate trigonometric functions. While they can give very accurate results, they are not always exact. The value of 2/√5 is a precise value, but the calculator's result for cos(22.5°) is an approximation.

## 4. Can cos(22.5°) be expressed as a decimal?

Yes, cos(22.5°) can be expressed as a decimal. The approximate value is 0.9239. However, it is important to note that this is an approximation and not the exact value.

## 5. How can we prove that cos(22.5°) is not equal to 2/√5?

To prove that cos(22.5°) is not equal to 2/√5, we can use the unit circle. The unit circle is a circle with a radius of 1 and is used to represent the values of trigonometric functions. The point on the unit circle corresponding to an angle of 22.5° has coordinates (√2/2, √2/2). Using the Pythagorean theorem, we can calculate the length of the hypotenuse of the right triangle formed by this point and the origin to be √2. Therefore, the ratio of the adjacent side to the hypotenuse is √2/√2, which simplifies to 1. This shows that the value of cos(22.5°) is 1, which is not equal to 2/√5.

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